Given the graph of f of x,
we're asked to graph g of x equals two times f
of the quantity x plus three.
The two indicates a vertical stretch by a factor of two
and the x plus three indicates a horizontal shift,
left three units.
For a quick review, when we have g of x equals a times f
of x, we have a vertical stretch
or compression, given a is greater than one.
If we have g of x equals a times f of x,
the graph has vertically stretched by a factor of a.
If we have g of x equals one divided by a times f of x,
then the graph is vertically compressed
by a factor of one divided by a.
If we have g of x equals f of the quantity x plus c,
the value of c indicates a shift left or right,
where if c is less than zero, or we have subtraction,
the graph has shifted right c units.
If c is greater than zero, or we have addition,
the graph is shifted left c units.
Notice how the shift left
or right might be in the opposite direction
than we think because if we have subtraction,
the graph has shifted right.
If we have addition, the graph is shifted left.
So going back to our example,
again, the two indicates a vertical stretch
by a factor of two
and the x plus three indicates a shift left three units.
Let's record this information.
To vertically stretch a graph by a factor of two,
we multiply the y-coordinates by two.
To shift a graph left three units,
we subtract three from the x-coordinates.
And because f of x is a segment,
we can perform these transformations on the two endpoints
to find the endpoints for the graph of g of x.
Notice how the endpoints
for f of x are negative two comma negative one,
and four comma two.
So starting
with the endpoint negative two comma negative one,
let's perform the vertical stretch by a factor
of two and a shift left three units.
So for the corresponding endpoint on g of x,
to shift the graph left three units,
we subtract three from the x-coordinate,
which gives us an x-coordinate of negative two minus three
and then to vertically stretch the graph by a factor of two,
we multiply the y-coordinate by two,
which gives us two times negative one for the y-coordinate.
Then therefore the corresponding endpoint
for g of x is negative five, comma negative two.
And I'll perform the same transformation
on the endpoint four comma two.
So to shift the graph left three units,
we subtract three from the x-coordinate,
which gives us four minus three.
To vertically stretch by a factor of two,
we multiply the y-coordinate by two,
which gives us two times two.
The corresponding endpoint on g of x is one comma four.
Let's go ahead and plot these two endpoints
and then sketch g of x.
So we have negative five comma negative two,
then we have one comma four.
Then therefore this segment is the graph of g of x.
Let's take a look at a second example.
Again, we're given the graph of f of x
and asked to graph g of x equals 1/3 times f
of the quantity x minus two.
So here the one 1/3 indicates a vertical compression
by a factor of 1/3
and the x minus two indicates a shift right two units.
Let's record this.
Let's find the endpoints for f of x.
We have negative five comma three.
Then we have three comma negative six.
So starting with the endpoint negative five comma three.
To vertically compress the graph by a factor of 1/3,
we multiply the y-coordinate by 1/3.
To shift the graph right two units,
we add two to the x-coordinate,
which means the corresponding endpoint on g of x
would have an x-coordinate of negative five plus two
and a y-coordinate of 1/3 times three.
Simplifying, we have negative three comma one.
Now using the other endpoint of three common negative six,
we perform the same transformation.
We shift the graph right two units.
We add two to the x-coordinate.
For your vertical compression by a factor of 1/3,
we multiply the y-coordinate by 1/3.
Simplifying, we have the ordered pair five comma,
1/3 times negative six is, negative two.
Let's go ahead and plot the endpoints for g of x
and then sketch g of x.
So we have negative three comma one.
Then we have five comma negative two.
And therefore this is the graph of g of x.
I hope you found this helpful.